Long Division with 2-Digit Divisors: Fifth Grade Mathematics
Dividing by 2-digit divisors requires estimating quotient digits, multiplying multi-digit values, and adjusting estimates when products are too large or differences exceed the divisor.
The Strategy for Estimating Quotient Digits
When dividing by a 2-digit number like 32, look at the leading digit of the divisor (3) and the leading digits of the dividend to estimate each quotient digit.
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| ESTIMATING A 2-DIGIT QUOTIENT DIGIT |
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| Problem: 864 / 32 |
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| Step 1: Divisor 32 has 2 digits. Look at first two digits of dividend: |
| Does 32 go into 86? Yes! |
| Step 2: Cover the ones digits of both numbers to estimate: |
| Cover 2 in 32 (think 3) and cover 6 in 86 (think 8). |
| How many 3s in 8? About 2. |
| Step 3: Test 2: 2 * 32 = 64. 64 <= 86, so 2 is correct! |
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Full Long Division Algorithm Walkthrough
Let us complete 864 / 32 step by step.
2 7
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3 2 | 8 6 4
- 6 4 <--- 2 * 32 = 64. Subtract: 86 - 64 = 22.
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2 2 4 <--- Bring down 4. Think: 3 into 22 is about 7.
- 2 2 4 <--- 7 * 32 = 224. Subtract: 224 - 224 = 0.
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0 <--- Quotient is 27 with remainder 0.
Adjusting Trial Quotient Digits
Estimating quotient digits is a trial process. Sometimes your estimated digit will be too large (the product exceeds the current working number) or too small (the subtracted difference is greater than or equal to the divisor).
Case 1: Estimate is Too Large
Working with 142 / 28:
You estimate 2 into 14 = 7.
Test: 7 * 28 = 196.
196 > 142! 7 is too large. Adjust down to 6 or 5.
Test 5: 5 * 28 = 140. 140 <= 142. Perfect!
Case 2: Estimate is Too Small
Working with 195 / 24:
You estimate 2 into 19 = 7.
Test: 7 * 24 = 168.
Subtract: 195 - 168 = 27.
27 >= 24 (divisor)! The difference is too big!
Adjust estimate up to 8. Test 8: 8 * 24 = 192.
Subtract: 195 - 192 = 3. 3 < 24. Perfect!
Formatting and Neatness Tips
Write each quotient digit directly above the place value digit that was just brought down. If you divide into 86, the first quotient digit sits directly above the 6. Placing digits in improper columns misaligns tens and ones.
Exercises and Solutions
Exercise 1: Standard 2-Digit Long Division. Solve 1,848 / 42 using the long division algorithm. Show each subtraction, bringing down, and quotient digit placement.
Solution 1: Divisor 42 has two digits. Compare to first two digits of dividend: 42 does not go into 18. Look at first three digits: 184. Estimate 4 into 18: about 4 times. Multiply: 4 * 42 = 168. Subtract: 184 - 168 = 16 (since 16 < 42, 4 is correct). Place 4 above the 4 in the dividend. Bring down 8 to make 168. Estimate 4 into 16: exactly 4 times. Multiply: 4 * 42 = 168. Subtract: 168 - 168 = 0. Remainder is 0. The final quotient is 44.
Exercise 2: Trial Adjustment Example. When solving 2,548 / 38, a student tests 7 as the first quotient digit for 254 / 38. Show whether 7 is correct, too large, or too small, and complete the division.
Solution 2: Test the student's trial digit 7 by multiplying 7 * 38 = 266. Compare 266 to the working number 254: 266 is greater than 254, which means 7 is too large. Adjust the quotient digit down to 6. Test 6: 6 * 38 = 228. Subtract 254 - 228 = 26. Since 26 is less than 38, 6 is correct. Place 6 above the 4 in the dividend. Bring down 8 to form 268. Estimate 38 into 268: test 7 (since 7 * 38 = 266). Subtract: 268 - 266 = 2. Since 2 is less than 38, 7 is correct. The final quotient is 67 with a remainder of 2 (67 R2).